Exact solutions, bifurcation analysis and chaotic behavior of high-order nonlinear Schrödinger equation with conformal fractional derivative

This study thoroughly investigates the dynamical properties, analytical solutions and chaotic behavior of the high-order nonlinear Schrödinger equation with conformal fractional derivative. After obtaining the dynamic properties by a third-order polynomial complete discriminant system, the study further utilizes a fourth-order polynomial complete discrimination system to derive all traveling wave solutions, including rational solutions, solitary wave solutions, triangular function periodic solutions, and Jacobian elliptic function periodic solutions. In addition, based on different phase diagrams and largest Lyapunov exponents, the chaotic behavior is numerically observed by considering different external perturbation terms and perturbation functions. The innovation of this work lies in using the same method to qualitatively and quantitatively analyze the proposed equation. The type of equation solution is predicted using the results of qualitative analysis, and the accuracy of the prediction can be verified through specific parameter values.